On a problem of Hajdu and Tengely

TL;DR

This paper completes the classification of primitive arithmetic progressions with squares and fifth powers, proving the only such progression is trivial, by analyzing rational points on hyperelliptic curves using descent and Chabauty's method.

Contribution

It proves the uniqueness of the trivial progression involving squares and fifth powers by applying advanced number theory techniques to hyperelliptic curves.

Findings

No nontrivial rational points on two hyperelliptic curves.

Identified generators for a subgroup of the Mordell-Weil group.

Confirmed the only rational points are the obvious trivial ones.

Abstract

We prove a result that finishes the study of primitive arithmetic progressions consisting of squares and fifth powers that was carried out by Hajdu and Tengely in a recent paper: The only arithmetic progression in coprime integers of the form (a^2, b^2, c^2, d^5) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generators for a subgroup of finite index of the Mordell-Weil group of the last curve. Applying Chabauty's method, we prove that the only rational points on this curve are the obvious ones.